Polymer Physics

resulting ina¼3/4. This gives

xC^3 =^4 (4.32)

and then

Rn^1 =^2 x^1 =^6 n^1 =^2 C^1 =^8 (4.33)

For polymer solutions with the coil size changing with polymer concentrations,
we normally call themsemi-dilute solutions. With a further increase ofC,xwill
eventually decrease down to the size of chain units, thenxno longer changes withC
whenC>C**. We finally reach

Rn^1 =^2 (4.34)

In summary, over the whole concentration range, we have

In dilute solutions,C<C,R/n3/5.
In semi-dilute solutions,C<C<C,R/n1/2C1/8.
In concentrated solutions,C>C,R/n1/2.

4.2.4 Single-Chain Conformation in Thermal Dilute Solutions

In dilute solutions, the single polymer coil expands in the athermal solvent. In a
good solvent, the coil will expand more significantly. In contrast, in a poor solvent,
the chain units and the solvent undergo a phase separation under a proper thermo-
dynamic condition. Consequently, the single chain will collapse drastically into a
condensed sphere. Therefore, the internal concentration reaches

Cint/

n

R^3

 1 (4.35)

and

Rn^1 =^3 (4.36)

with the scaling exponent ofn¼0.33
With the change of temperature or with the addition of a poor solvent, the
transition of the single chain from an expanded coil to a hard sphere is called
collapse transition. Meanwhile, the scaling exponent of the coil size reduces from
0.6 to 0.33. Therefore,n¼0.5 exists during the collapse transition, representing a
scaling relationship for ideal coils. We defined the thermodynamic condition for
this transient pseudo-ideal state as thetheta point.

4.2 Single-Chain Conformation in Polymer Solutions 55